June29,2013,AlbaIulia F.-H.Vasilescu NormalExtensionsofLinearRelationsviaQuaternionicCayleyTransforms

Normal Extensions of Linear Relations via Quaternionic Cayley Transforms

F.-H. Vasilescu

Department of Mathematics University of Lille 1

June 29, 2013, Alba Iulia

Outline

1 Introduction

Notation and Preliminaries Normal Relations

2 Quaternionic Cayley Transforms of Linear Relations E–Cayley Transform

InverseE–Cayley Transform

3 Unitary Operators and the Inverse Quaternionic Cayley Transform

Special Unitary Operators

Other special Classes of Operator and Relations

4 Normal Extensions Subnormality

Abstract

In a previous work, the author has introduced a notion of quaternionic Cayley transform, valid for some classes of pairs of symmetric operators. In a second work, the former definition was slightly modified, leading to more direct proofs, using von Neumann’s Cayley transform.

In the present work, written in cooperation withAdrian Sandovici, a quaternionic Cayley transform for linear relations is introduced and some of its properties are exhibited. We emphasize the role played by the linear relations whose quaternionic Cayley transforms are unitary operators, which happen to be normal relations, and investigate the class of those linear relations which extend

Abstract

In a previous work, the author has introduced a notion of quaternionic Cayley transform, valid for some classes of pairs of symmetric operators. In a second work, the former definition was slightly modified, leading to more direct proofs, using von Neumann’s Cayley transform.

In the present work, written in cooperation withAdrian Sandovici, a quaternionic Cayley transform for linear relations is introduced and some of its properties are exhibited. We emphasize the role played by the linear relations whose quaternionic Cayley transforms are unitary operators, which happen to be normal relations, and investigate the class of those linear relations which extend

Some References

R. Arens,Operational calculus of linear relations, Pacific J.

Math. 11 (1961) 9-23.

E. A. Coddington,Extension theory of formally normal and symmetric subspaces, Mem. Amer. Math. Soc., 134, 1973.

A. Sandovici and F.-H. Vasilescu,Normal Extensions of Subnormal Linear Relations via Quaternionic Cayley Transforms, Monatsh. Math. (to appear)

F.-H. Vasilescu,Quaternionic Cayley transform, J.

Functional Analysis164(1999), 134-162.

F.-H. Vasilescu,Quaternionic Cayley transform revisited,

Introduction

Let(H,h·,·i)be a complex Hilbert space and letU be a unitary operator acting inH. IfIis the identity operator onH, set

S={{(I−U)x,i(I+U)x};x ∈ H}, (1) which is a linear subspace ofH²=H × H. The spaceSis the graph of a linear transformation inHif and only if the operator I−U is injective. In this case, the corresponding linear

transformation is known as theinverse Cayley transformof the unitary operatorU, which is, in general, a (not necessarily bounded) self-adjoint operator.

Nevertheless, the spaceSgiven by (1) is well defined without the conditionI−U invertible, and it is what is called as alinear relationinH(followingArens). Moreover, we have

S^∗:={{u,v} ∈ H²:hx,vi=hy,ui, ∀ {x,y} ∈S}=S, whereS^∗ stands for the adjoint relation ofS. In other words,S is aself-adjointlinear relation (formal definitions will be later given).

We shall define a Cayley transform for some linear relations.

Briefly, considering a linear relationSinH²(that is, a linear subspace ofH²× H²), we associate it with the linear relation

V ={{{x₁⁰,x₂⁰}+{ix₁,−ix₂},{x₁⁰,x₂⁰}+{−ix₁,ix₂}} : {{x₁,x₁⁰},{x₂,x₂⁰}} ∈S}, which is a quaternionic type Cayley transform ofS. Under some natural conditions, the linear spaceV is the graph of a partial isometry inH². Our aim is to study various properties relating the linear relationSand the partial isometryV.

An "extreme" situation is whenU is an operator of the form U =

T iA iA T^∗

,

withT normal andAself-adjoint inH, such thatTT^∗+A²=I andAT =TA.In this case,U is unitary, and it is a quaternionic Cayley transform of the linear relation

{{i(T−I)x−Ay,Ax−i(T^∗−I)y},(T+I)x+iAy,iAx+ (T^∗+I)y}, {x,y} ∈ H²},

which is normal (in a sense to be defined).

Notation and Preliminaries

LetHbe a complex Hilbert space. Alinear relation(briefly, alr) inHis a vector subspace ofH²=H × H. The elements ofH² are represented as pairs{x,y}, withx,y ∈ H.

For alr T inH, we denote byD(T),R(T),N(T),M(T)its domain of definition, itsrange, itskernel, and itsmultivalued part, respectively.

We often identify a linear mapS:D(S)⊂ H 7→ Hwith its graph G(S).

Notation and Preliminaries

LetHbe a complex Hilbert space. Alinear relation(briefly, alr) inHis a vector subspace ofH²=H × H. The elements ofH² are represented as pairs{x,y}, withx,y ∈ H.

For alr T inH, we denote byD(T),R(T),N(T),M(T)its domain of definition, itsrange, itskernel, and itsmultivalued part, respectively.

We often identify a linear mapS:D(S)⊂ H 7→ Hwith its graph G(S).

IfS,T arelr inH², theircompositionis given by ST ={{x,z};∃y :{x,y} ∈T,{y,z} ∈S}.

Theadjoint T^∗ ofT is given by

T^∗ ={{u,v}:hu,yi=hv,xi;∀{x,y} ∈T}.

IfT ⊂T^∗,T is said to besymmetric.

IfT =T^∗,T is said to beself-adjoint.

IfS,T arelr inH², theircompositionis given by ST ={{x,z};∃y :{x,y} ∈T,{y,z} ∈S}.

Theadjoint T^∗ ofT is given by

T^∗ ={{u,v}:hu,yi=hv,xi;∀{x,y} ∈T}.

IfT ⊂T^∗,T is said to besymmetric.

IfT =T^∗,T is said to beself-adjoint.

IfS,T arelr inH², theircompositionis given by ST ={{x,z};∃y :{x,y} ∈T,{y,z} ∈S}.

Theadjoint T^∗ ofT is given by

T^∗ ={{u,v}:hu,yi=hv,xi;∀{x,y} ∈T}.

IfT ⊂T^∗,T is said to besymmetric.

IfT =T^∗,T is said to beself-adjoint.

IfS,T arelr inH², theircompositionis given by ST ={{x,z};∃y :{x,y} ∈T,{y,z} ∈S}.

Theadjoint T^∗ ofT is given by

T^∗ ={{u,v}:hu,yi=hv,xi;∀{x,y} ∈T}.

IfT ⊂T^∗,T is said to besymmetric.

IfT =T^∗,T is said to beself-adjoint.

Normal Relations

A linear relationSin a Hilbert spaceHis said to beformally normal(Coddington) if there exists an isometryV :S →S^∗ of the form

V{x,x⁰}={x,x⁰⁰}, {x,x⁰} ∈S, {x,x⁰⁰} ∈S^∗. (2) In particular,kx⁰k=kx⁰⁰k.

A formally normal linear relationS inHis said to benormal (Coddington) if the isometryV is surjective.

Normal relations are automatically closed.

Normal Relations

A linear relationSin a Hilbert spaceHis said to beformally normal(Coddington) if there exists an isometryV :S →S^∗ of the form

V{x,x⁰}={x,x⁰⁰}, {x,x⁰} ∈S, {x,x⁰⁰} ∈S^∗. (2) In particular,kx⁰k=kx⁰⁰k.

A formally normal linear relationS inHis said to benormal (Coddington) if the isometryV is surjective.

Normal relations are automatically closed.

Normal Relations

A linear relationSin a Hilbert spaceHis said to beformally normal(Coddington) if there exists an isometryV :S →S^∗ of the form

V{x,x⁰}={x,x⁰⁰}, {x,x⁰} ∈S, {x,x⁰⁰} ∈S^∗. (2) In particular,kx⁰k=kx⁰⁰k.

A formally normal linear relationS inHis said to benormal (Coddington) if the isometryV is surjective.

Normal relations are automatically closed.

Characterization of Normality

As in the case of (unbounded) normal operators, we have the following:

Theorem (Sandovici+V)

LetSbe a linear relation in a Hilbert spaceH. Equivalent are:

1 Sis a normal linear relation inH;

2 Sis closed,D(S) =D(S^∗), andS^∗S=SS^∗.

Quaternionic Cayley Transform of Linear Relations

LetSbe a symmetric linear relation inH. We define the relation V ={{x⁰+ix,x⁰−ix};{x,x⁰} ∈S},

which is called theCayley transformofS(Arens).

BecauseSis symmetric, we have

kx⁰±ixk²=kxk²+kx⁰k², {x,x⁰} ∈S.

Therefore,V is (the graph of) an isometry, defined on D(V) ={x⁰+ix,;{x,x⁰} ∈S} ⊂ H, whose range is R(V) ={x⁰−ix;{x,x⁰} ∈S} ⊂ H.

Quaternionic Cayley Transform of Linear Relations

LetSbe a symmetric linear relation inH. We define the relation V ={{x⁰+ix,x⁰−ix};{x,x⁰} ∈S},

which is called theCayley transformofS(Arens).

BecauseSis symmetric, we have

kx⁰±ixk²=kxk²+kx⁰k², {x,x⁰} ∈S.

Therefore,V is (the graph of) an isometry, defined on D(V) ={x⁰+ix,;{x,x⁰} ∈S} ⊂ H, whose range is R(V) ={x⁰−ix;{x,x⁰} ∈S} ⊂ H.

Version of a von Neumann’s Theorem

The next result was proved byArens:

TheoremLetS be a symmetric relation in the Hilbert spaceH.

The relationS is self-adjoint if and only if the Cayley transform ofSis a unitary operator inH.

Recall: Algebra of Quaternions

Consider the 2×2-matrices I=

1 0 0 1

, J=

1 0 0 −1

,

K=

0 1

−1 0

, L=

0 1 1 0

.

The Hamilton algebra of quaternionsHcan be identified with theR-subalgebra of the algebraM2of 2×2-matrices with complex entries, generated by the matricesI,iJ,KandiL. This allows us to regard the elements ofHas matrices and to perform some operations inM2rather than inH.

Recall: Some Elementary Operations

We haveJ^∗=J,K^∗ =−K,L^∗=L,J²=−K²=L²=I, JK=L=−KJ,KL=J=−LK,JL=K=−LJ,where the adjoints are computed in the Hilbert spaceC².

We also putE=iJ,F=iL, and we we haveE^∗ =−E,E²=−I, F^∗=−F,F²=−I

A Preliminary Lemma

LetHbe a complex Hilbert space. The matrices fromM2

clearly act onH²(whose natural norm is denoted byk ∗ k₂).

Lemma

LetSbe a linear relation inH². Suppose that the linear relation JSis symmetric. Then we have

kx⁰±Exk²₂=kx⁰k²₂+kxk²₂, {x,x⁰} ∈S. (3)

E–Cayley Transform

LetSbe a linear relation inH²such thatJS is symmetric.

E–Cayley transform ofS:

V :=

{x⁰+Ex,x⁰−Ex} : {x,x⁰} ∈S . (4) As we havekx⁰+Exk₂=kx⁰−Exk₂, it follows thatV is (the graph of) a partial isometry withD(V) =R(S+E)and R(V) =R(S−E).

In fact,V given by

V :R(S+E)7→R(S−E), V(x⁰+Ex) =x⁰−Ex, {x,x⁰} ∈S.

E–Cayley Transform

LetSbe a linear relation inH²such thatJS is symmetric.

E–Cayley transform ofS:

V :=

{x⁰+Ex,x⁰−Ex} : {x,x⁰} ∈S . (4) As we havekx⁰+Exk₂=kx⁰−Exk₂, it follows thatV is (the graph of) a partial isometry withD(V) =R(S+E)and R(V) =R(S−E).

In fact,V given by

V :R(S+E)7→R(S−E), V(x⁰+Ex) =x⁰−Ex, {x,x⁰} ∈S.

E–Cayley Transform

LetSbe a linear relation inH²such thatJS is symmetric.

E–Cayley transform ofS:

V :=

{x⁰+Ex,x⁰−Ex} : {x,x⁰} ∈S . (4) As we havekx⁰+Exk₂=kx⁰−Exk₂, it follows thatV is (the graph of) a partial isometry withD(V) =R(S+E)and R(V) =R(S−E).

In fact,V given by

V :R(S+E)7→R(S−E), V(x⁰+Ex) =x⁰−Ex, {x,x⁰} ∈S.

F–Cayley Transform

IfLSis symmetric, a similar discussion leads to the definition of an operatorW given by

W :R(S+F)7→R(S−F), W(x⁰+Fx) =x⁰−Fx, {x,x⁰} ∈S, which is again a partial isometry. The operatorW is called the F–Cayley transformofS.

Because the two Cayley transforms defined above are alike, in the sequel we shall mainly deal with theE–Cayley transform.

Properties of the E–Cayley Transform

Lemma

LetSbe a linear relation inH²such thatJS is symmetric, and letV be theE–Cayley transform ofS. We have the following:

(a)V is closed if and only ifSis closed, and if and only if the spacesR(S±E)are closed;

(b)One hasN(I−V) =M(S). The operatorI−V is injective if and only ifSis an operator. Moreover,Sis densely defined if and only if the spaceR(I−V)is dense inH²;

(c)IfSK⊂KS, thenSK=KSandV⁻¹=−KVK;

(d)The lrJSis self-adjoint if and only ifV is unitary inH². (e)TheE–Cayley transform is an inclusion preserving map.

Inverse E–Cayley Transform

LetV :D(V)⊂ H²7→ H²be a partial isometry, for which we define the following:

Inverse E–Cayley transform:

S :={{E(V −I)x,(V +I)x} : x ∈D(V)}, which is a linear relation inH².

In a similar way, we can define theinverseF–Cayley transform.

These two (quaternionic) inverse Cayley transforms have similar properties, and so we shall deal only with the inverse E–Cayley transform.

Inverse E–Cayley Transform

LetV :D(V)⊂ H²7→ H²be a partial isometry, for which we define the following:

Inverse E–Cayley transform:

S :={{E(V −I)x,(V +I)x} : x ∈D(V)}, which is a linear relation inH².

In a similar way, we can define theinverseF–Cayley transform.

These two (quaternionic) inverse Cayley transforms have similar properties, and so we shall deal only with the inverse E–Cayley transform.

Properties of the Inverse E–Cayley Transform

LemmaLetV :D(V)⊂ H²7→ H²be a partial isometry. Then the linear relation

S={{E(V−I)x,(V +I)x} : x ∈D(V)}. has the following properties:

(i)the linear relationJSis symmetric and theE–Cayley transform ofSisV;

(ii)we haveV⁻¹=−KVKif and only ifSK=KS.

Global Properties

We summarize the properties of the quaternionic Cayley transform:

Theorem

TheE-Cayley transform is an inclusion preserving bijective map assigning to each linear relationSinH²withJS symmetric a partial isometryV inH². Moreover:

(1)the operatorV is closed if and only if the linear relationSis closed;

(2)the equalityV⁻¹=−KVKholds if and only if the equality SK=KSholds;

(3)the linear relationJSis self-adjoint if and only ifV is unitary onH².

Special Unitary Operators

In this this section we are particularly interested in those unitary operators producing normal linear relations (in particular

(unbounded) normal operators), via the inverseE–Cayley transform.

LemmaLetUbe a bounded operator onH². The operatorUis unitary and has the propertyU^∗ =−KUKif and only if there are a bounded operatorT and bounded self-adjoint operators A,B onHsuch thatTT^∗+A²=I,T^∗T +B²=I,AT =TBand

U=

T iA iB T^∗

, whereIthe identity onH.

Some Auxilliary Results

LemmaLetUbe a unitary operator onH²withU^∗ =−KUK. If we set

S={{E(U−I)x,(U+I)x} : x ∈ H}. we have thatSis a closed linear relation and

S^∗ ={{(I−U)x,E(I+U)x} : x ∈ H}.

LemmaLetUbe an operator onH²having the form U=

T iA iB T^∗

,

withT,A=A^∗,B=B^∗ bounded operators onH, such that

∗ 2 ∗ 2

Intertwining Isometries

LemmaLetV be a partial isometry such thatV⁻¹=−KVK.

LetSbe the inverseE–Cayley transform ofV. Equivalent are:

(i)JD(S)⊂D(S)and there exists an isometryH :S→Sof the formH{x,x⁰}={Jx,x⁰⁰};

(ii)there exists an isometryG:D(V)7→D(V)such that E(I−V) = (I−V)G.

Remark

(1) The isometryGgiven by the previous Lemma is not uniquely determined. IfG₁,G₂are two isometries as in this lemma, then the operatorG₁−G₂is clearly defined onD(V) and has values in the kernel ofI−V.

(2) Assume that the isometryHin the previous Lemma is surjective. Then it can be shown that the isometric operatorG may be chosen to be surjective. Conversely, ifGis surjective, the operatorH may be also chosen to be surjective.

Proposition

LetUbe a unitary operator onH²with the property

U^∗=−KUK. Let alsoSbe the inverseE–Cayley transform of U. The linear relationSis normal if and only is there exists a unitary operatorG_U onH²such thatE(I−U) = (I−U)G_U and (G_U)^∗ =−G_U.

Main result and the Class U

(H

)

Theorem

LetUbe a unitary operator onH²with the property

U^∗=−KUK, and letS be the inverseE–Cayley transform ofU.

The linear relationSis normal if and only if (U+U^∗)E=E(U+U^∗).

DefinitionLetU(H²)be the set of all unitary operators inH². We also set

U_C(H²) ={U ∈ U(H²);U^∗=−KUK, (U+U^∗)E=E(U+U^∗)},

Main result and the Class U

(H

)

Theorem

LetUbe a unitary operator onH²with the property

U^∗=−KUK, and letS be the inverseE–Cayley transform ofU.

The linear relationSis normal if and only if (U+U^∗)E=E(U+U^∗).

DefinitionLetU(H²)be the set of all unitary operators inH². We also set

U_C(H²) ={U ∈ U(H²);U^∗=−KUK, (U+U^∗)E=E(U+U^∗)},

The Class N

(H

)

DefinitionWe introduce the following class of linear relations in H²:

N_IC(H²) ={S ⊂ H²× H²; S normal,(JS)^∗ =JS,KS=SK}.

RemarkThe previous results show that the map

N_IC(H²)3S7→U_S ∈ U_C(H²),

whereU_S stands for theE–Cayley transform ofS, is bijective.

The Class N

(H

)

DefinitionWe introduce the following class of linear relations in H²:

N_IC(H²) ={S ⊂ H²× H²; S normal,(JS)^∗ =JS,KS=SK}.

RemarkThe previous results show that the map N_IC(H²)3S7→U_S ∈ U_C(H²),

whereU_S stands for theE–Cayley transform ofS, is bijective.

Subnormality

Adapting the terminology from operator theory, we may say that a linear relationSin a Hilbert spaceHissubnormalif there exists a Hilbert spaceK ⊃ Hand a normal relationN inK such thatS ⊂N. Nevertheless, we are particularly interested to solve such a problem with the restrictionK=H. In fact, to apply the quaternionic Cayley transform machinery, we consider only linear relations in the Hilbert spaceH².

The Class S

(H

)

We start by introducing a class of linear relations having necessary properties connected to subnormality.

LetT :be a linear relation inH², withD(T) =D₀⊕D₀,D₀⊂ H, which is equivalent to the inclusions

(i)JD(T)⊂D(T)andKD(T)⊂D(T).

Furthermore, assume thatT satisfies the following conditions:

(ii)JT is symmetric;

(iii)TK=KT;

(iv)there exists a surjective isometryH_T :T →T of the form H_T{x,x⁰}={Jx,x⁰⁰}; for all{x,x⁰} ∈T.

Denote byS_IC(H²)the set of those linear relationsT inH² such that(i)–(iv)hold.

The Class P

(H

)

LetP_C(H²)be the set of those partial isometries V :D(V)⊂ H²7→ H²such that:

(a)V⁻¹=−KVK;

(b)ER(I−V) =R(I−V);

(c)there exists a surjective isometryG:D(V)7→D(V)such thatE(I−V) = (I−V)G.

It was proved that theE–Cayley transform is a bijective map fromS_IC(H²)ontoP_C(H²). Note also thatU_C(H²)⊂ P_C(H²)by some of the previous results.

Normal Extensions

A decomposition result:

PropositionLetU ∈ U_C(H²)and letD ⊂ H²be a closed subspace with the propertiesKU(D)⊂ Dand

E(I−U)(D)⊂(I−U)(D). IfV =U|D,E =D^⊥andW =U|E, thenU=V⊕W andV,W ∈ P_C(H²).

Main extension result:

Theorem

LetT ∈ S_IC(H²). Equivalent are:

(i)the linear relationT has an extensionS inN_IC(H²)such thatH_T =H_S|T;

(ii)there exists aW ∈ P_C(H²), withD(W) =R(T +E)^⊥.

Normal Extensions

A decomposition result:

PropositionLetU ∈ U_C(H²)and letD ⊂ H²be a closed subspace with the propertiesKU(D)⊂ Dand

E(I−U)(D)⊂(I−U)(D). IfV =U|D,E =D^⊥andW =U|E, thenU=V⊕W andV,W ∈ P_C(H²).

Main extension result:

Theorem

LetT ∈ S_IC(H²). Equivalent are:

(i)the linear relationT has an extensionS inN_IC(H²)such thatH_T =H_S|T;

(ii)there exists aW ∈ P_C(H²), withD(W) =R(T +E)^⊥.

The next assertion provides an extension result for linear relations. In particular, this includes the case of not necessarily densely defined operators.

Corollary

LetT ∈ S_IC(H²)be closed and letV be theE–Cayley

transform ofT. Then the linear relationT has an extension in N_IC(H²)if and only if there exists aW ∈ P_C(H²), with the propertyD(W) =R(T +E)^⊥.

Thank you for your attention!